The Split Attractor Flow Conjecture, originally proposed in [1]}, [2]}
and sharpened in [3]}, amounts to the statement that for any \(z\in \Pi \)  and \(\gamma \in \Gamma \) , there exists only a finite number
of decompositions \(\gamma =\sum _i \gamma _i\)  such that \(\mathcal {T}_z(\lbrace \gamma _i\rbrace )\)  is non-empty; the rational index \(\bar{\Omega }_z(\gamma )\)  is then obtained by summing over all
attractor flow trees,
\(\bar{\Omega }_z(\gamma ) = \sum _{\gamma =\sum _i \gamma _i}\frac{1}{\vert \operatorname{Aut}(\lbrace \gamma _i\rbrace )\vert }\sum _{T\in \mathcal {T}_z(\lbrace \gamma _i\rbrace )} g_z(T)\,\prod _{i} \bar{\Omega }_\star (\gamma _i)\) 
